Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj873 Structured version   Visualization version   GIF version

Theorem bnj873 32306
Description: Technical lemma for bnj69 32392. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj873.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj873.7 (𝜑′[𝑔 / 𝑓]𝜑)
bnj873.8 (𝜓′[𝑔 / 𝑓]𝜓)
Assertion
Ref Expression
bnj873 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Distinct variable groups:   𝐷,𝑓,𝑔   𝑓,𝑛,𝑔   𝜑,𝑔   𝜓,𝑔
Allowed substitution hints:   𝜑(𝑓,𝑛)   𝜓(𝑓,𝑛)   𝐵(𝑓,𝑔,𝑛)   𝐷(𝑛)   𝜑′(𝑓,𝑔,𝑛)   𝜓′(𝑓,𝑔,𝑛)

Proof of Theorem bnj873
StepHypRef Expression
1 bnj873.4 . 2 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
2 nfv 1915 . . 3 𝑔𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
3 nfcv 2955 . . . 4 𝑓𝐷
4 nfv 1915 . . . . 5 𝑓 𝑔 Fn 𝑛
5 bnj873.7 . . . . . 6 (𝜑′[𝑔 / 𝑓]𝜑)
6 nfsbc1v 3740 . . . . . 6 𝑓[𝑔 / 𝑓]𝜑
75, 6nfxfr 1854 . . . . 5 𝑓𝜑′
8 bnj873.8 . . . . . 6 (𝜓′[𝑔 / 𝑓]𝜓)
9 nfsbc1v 3740 . . . . . 6 𝑓[𝑔 / 𝑓]𝜓
108, 9nfxfr 1854 . . . . 5 𝑓𝜓′
114, 7, 10nf3an 1902 . . . 4 𝑓(𝑔 Fn 𝑛𝜑′𝜓′)
123, 11nfrex 3268 . . 3 𝑓𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)
13 fneq1 6414 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑛𝑔 Fn 𝑛))
14 sbceq1a 3731 . . . . . 6 (𝑓 = 𝑔 → (𝜑[𝑔 / 𝑓]𝜑))
1514, 5syl6bbr 292 . . . . 5 (𝑓 = 𝑔 → (𝜑𝜑′))
16 sbceq1a 3731 . . . . . 6 (𝑓 = 𝑔 → (𝜓[𝑔 / 𝑓]𝜓))
1716, 8syl6bbr 292 . . . . 5 (𝑓 = 𝑔 → (𝜓𝜓′))
1813, 15, 173anbi123d 1433 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′)))
1918rexbidv 3256 . . 3 (𝑓 = 𝑔 → (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)))
202, 12, 19cbvabw 2867 . 2 {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)} = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
211, 20eqtri 2821 1 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Colors of variables: wff setvar class
Syntax hints:  wb 209  w3a 1084   = wceq 1538  {cab 2776  wrex 3107  [wsbc 3720   Fn wfn 6319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-fun 6326  df-fn 6327
This theorem is referenced by:  bnj849  32307  bnj893  32310
  Copyright terms: Public domain W3C validator